Let p be an odd prime. Given:
(x+1)^p = x^p + 1^p (mod p)
holds for any integer x, prove n^p = n(mod p) for all integers n.
Any help would be greatly appreciated!!
By the way you said in your hypothesis that p has to be odd, implying that this does not work for p=2. It does in fact work for p=2.
Also, the proof of the hypothesis is far more interesting than the proof of your statement, which is practically a corollary of the hypothesis.